Weak order on complete quadrics (Q2855928)

Let \(G\) be a semi-simple algebraic group over an algebraically closed field of characteristic different from 2 and let \(B\subset G\) be a Borel subgroup. A \(G\)-variety \(X\) is spherical if \(X\) has finitely many \(B\)-orbits. There are two natural orders on the finite set \(B(X)\) of \(B\)-stable subvarieties of \(X\):NEWLINENEWLINE I) The Bruhat-Chevalley order, which is defined by \(Y_1 \leq_B Y_2\Leftrightarrow Y_1\subset Y_2\). In the case of a smooth quadric, the combinatorics of this order was studied in [\textit{F. Incitti}, J. Algebr. Comb. 20, No. 3, 243--261 (2004; Zbl 1057.05079)].NEWLINENEWLINEII) The weak order \(\leq\): \( Y_1\) is covered by \(Y_2\) if and only if \(Y_2 = PY_1\) for some minimal parabolic subgroup \(P\subset G\). This order is weaker than the Bruhat-Chevalley order; moreover, given two \(B\)-orbit \(O_1\), \(O_2\) such that \(\overline{O}_1 \leq \overline{O}_2\), then \(O_1\), \(O_2\) lie in the same \(G\)-orbit. Thus, it suffices to study the weak order for homogeneous varieties. For varieties that are not homogeneous, information about the Bruhat-Chevalley order can be recovered from the knowledge of the weak order of each of the \(G\)-orbits.NEWLINENEWLINEIf \(X\) is homogeneous, then \(X\) is the unique maximal element of \(B(X)\). For \(Y\in B(X)\), the \(W\)-set of \(Y\), \(W(Y)\), consists of all \(w\in W\) of length \(\mathrm{codim}(Y)\), such that for some reduced expression \(w = s_1s_2\cdots s_l\) holds \(X = P_{s_1}P_{s_2}\cdots P_{s_l}Y\). If \(B(X)\) has a unique minimal element \(Y_0\), then the maximal chains in \(B(X)\) correspond to the reduced expressions of elements in \(W(Y_0)\). The principal result of this paper is the description of the \(W\)-sets for the unique minimal element in \(B(O)\) for each \(G\)-orbit \(O\) of the variety of complete quadrics \(M\). \(M\) is defined as the unique wonderful completion of \(\mathrm{PSL}_n/\mathrm{PSO}_n\) (see [\textit{C. De Concini} and \textit{C. Procesi}, Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]). In particular, the authors explicitly determine maximal chains in \(B(O)\). The boundary of \(M\) is a union of \(n-1\) smooth \(G\)-stable divisors with smooth transversal intersections. Each \(G\)-orbit in \(M\) corresponds to a subset of \(\{1, 2, \ldots, n - 1\}\); they are also in one-to-one correspondance with the set of compositions \(\mu\) of \(n\). Recall that a composition of \(n\) is an ordered sequence of positive integers that sum to \(n\). The \(B\)-orbits in \(O_\mu\) are parameterized by combinatorial objects that the authors call \(\mu\)-involutions. In the case of the open \(G\)-orbit, they are the involutions of the group \(S_n\) of permutations on \(n\) elements. To prove the main result, the author consider the Richardson-Springer monoid \(M(S_n)\) and define an action of \(M(S_n)\) over the set of \(\mu\)-involutions, such that \(O_{\mu'}\leq O_{\mu} \Leftrightarrow \mu'= w \cdot \mu\) for some \(w\in M(S_n)\).NEWLINENEWLINENEWLINENEWLINE Let \(X_\mu\) be the closure of the dense \(B\)-orbit of \(O_\mu\), \(Y_\mu\) the closure of the unique closed \(B\)-orbit of \(O_\mu\) and \(i : G/B^-\rightarrow X_\mu\) the natural inclusion. Finally, the authors determine \(i^*([Y_\mu]) \in H^*(G/B^-,\mathbb{Z})\).